Regular hypermaps
type: Item
instance of: scholarly article
title: Regular hypermaps
zbMATH Open document ID: 0665.57002
DOI: 10.1016/S0195-6698(88)80064-7
author: David Corn
author: David Singerman
published in: European Journal of Combinatorics
publication date: 1988-01-01T00:00:00Z
review text: This paper studies the correspondence between topological hypermaps and algebraic hypermaps. A topological map is a decomposition of a surface into polygonal cells. A topological hypermap is a similar decomposition of an orientable surface with the ``edges'' and ``vertices'' (hyperedges and hypervertices) homeomorphic to closed discs. More formally it is a triple \({\mathcal H}=(X,S,A)\) where X is an orientable surface, S and A have components (hypervertices and hyperedges) homeomorphic to closed discs, \(B=S\cap A\) (bits) is a finite set for X compact or satisfies some locally finite conditions for noncompact X, and the components (hyperfaces) of \(X\setminus (S\cup A)\) are homeomorphic to open discs. \({\mathcal H}\) is of type (\(\ell,m,n)\) if \(\ell\), m, and 2n are the least common multiples of the number of bits on the hypervertices, hyperedges and hyperfaces. The genus of \({\mathcal H}\) is the genus of X. An algebraic hypermap is a quadruple \({\mathcal A}=(G,B,\sigma,\alpha)\) where B is a set and \(\sigma\), \(\alpha\) are permutations of B such that \(G=gp<\sigma,\alpha >\) is transitive on B. The type of \({\mathcal A}\) is (\(\ell,m,n)\) where \(\ell\), m and n are the orders of \(\sigma\), \(\alpha\), and \(\sigma\) \(\alpha\) respectively. The genus, g, of \({\mathcal A}\) is given by \(z(\sigma)+z(\alpha)+z(\sigma \alpha)=| B| +2-2g\) where z(\(\zeta)\) is the number of cycles in the permutation \(\zeta\). Given a topological hypermap \({\mathcal H}\), an algebraic hypermap, Alg(\({\mathcal H})\), is constructed by letting B be the set of bits and \(\sigma\) (resp. \(\alpha)\) the permutation of B whose cycles are obtained by going around the hypervertices (resp. the hyperedges) in a positive sense. The permutation \(\sigma\) \(\alpha\) consists of cycles going around hyperfaces in a negative direction traversing two edges at a time (one from a hyperedge and one from a hypervertex). The construction of Alg(\({\mathcal H})\) preserves type and ge…
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 54B15
zbMATH DE Number: 4088513
zbMATH Keywords: decomposition of an orientable surface with the edges and vertices homeomorphic to closed discs
zbMATH Keywords: topological hypermaps
zbMATH Keywords: algebraic hypermaps
zbMATH Keywords: hypervertices
zbMATH Keywords: hyperedges
zbMATH Keywords: hyperfaces
zbMATH Keywords: genus
zbMATH Keywords: universal topological hypermap
zbMATH Keywords: universal algebraic hypermap
reviewed by: George E. Lang jun.
MaRDI profile type: Publication
cites work: Q4071773
cites work: On the automorphism group of a planar hypermap
cites work: Theory of Maps on Orientable Surfaces
cites work: The Riemann-Hurwitz formula for the centralizer of a pair of permutations
cites work: Finitely Maximal Fuchsian Groups
cites work: Automorphisms of Maps, Permutation Groups and Riemann Surfaces
cites work: Klein's Riemann Surface of Genus 3 and Regular Imbeddings of Finite Projective Planes
cites work: Hypermaps versus bipartite maps
cites work: Q4156452
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description: scientific article; zbMATH DE number 4088513